My question is, what if she is a widow and nobody is ready to take responsibility for her? I want to take responsibility for her with permission of my wife. Under this situation can I do nikah with her?

2 Answers
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All Praise to Allah Swt and blessings on Prophet Muhammad peace be upon him His family and companions

Whether she is a widow or you get permission from your wife you are still not allowed to marry her in this situation.

Prohibited to you [for marriage] are your mothers, your daughters, your sisters, your father's sisters, your mother's sisters, your brother's daughters, your sister's daughters, your [milk] mothers who nursed you, your sisters through nursing, your wives' mothers, and your step-daughters under your guardianship [born] of your wives unto whom you have gone in. But if you have not gone in unto them, there is no sin upon you. And [also prohibited are] the wives of your sons who are from your [own] loins, and that you take [in marriage] two sisters simultaneously, except for what has already occurred. Indeed, Allah is ever Forgiving and Merciful.(Quran 4:23)

Narrated Um Habiba: I said, "O Allah's Messenger (ﷺ)! Do you like to
have (my sister) the daughter of Abu Sufyan?" The Prophet (ﷺ) said,
"What shall I do (with her)?" I said, "Marry her." He said, "Do you
like that?" I said, "(Yes), for even now I am not your only wife, so I
like that my sister should share you with me." He said, "She is not
lawful for me (to marry)." I said, "We have heard that you want to
marry." He said, "The daughter of Um Salama?" I said, "Yes." He said,
"Even if she were not my stepdaughter, she should be unlawful for me
to marry, for Thuwaiba suckled me and her father (Abu Salama). So you
should neither present your daughters, nor your sisters, to
me."(Bukhari)

As long as you are married to your wife you can't marry her sister as Verse (4:23) clearly say

"Prohibited to you ... and that you take [in marriage] two sisters simultaneously"

And the sunna of the Prophet (peace be upon him) added also the aunts (paternal and maternal) of the wife as woman who are haram to marry simultaneously to your wife!

So one could only marry one of those if the wife died or one has divorced her.

Therefore: no matter what your wife says or permit you can't marry her sister (while you are married to her)!

But one thing must be added about "woman who are haram to marry simultaneously to your wife": they are non-mahram that means a man isn't allowed to look at such a woman (like the sister of the wife) unveiled! And she isn't allowed to be unveiled in front of him. As once the condition of marriage to her sister (niece in case of an aunt of the wife) would expire the man could marry her! So please notice that any of the women who are haram to marry simultaneously to your wife are no mahram!

But this doesn't mean you can't take responsibility for her or care for her. And she could stay with you and your wife under the conditions that you treat her like you would treat a foreign woman! This means never see her unveiled or never be alone with her!